Modeling the Winter-Tick Epizootic

By: Abbigail Ramnarine

Charlotte Beckford is a senior at Fordham College at Rose Hill, double majoring in mathematics with a concentration in applied mathematics and Spanish language and literature.  With a focus on the application of mathematics to the environment, Beckford began to explore the population dynamics regarding moose and winter-ticks and furthered her research as she enrolled in an Independent Study course. Through the use of models and theorizations, this project’s ultimate goal was to determine what actions could be taken to protect vulnerable moose populations from the deleterious effects of ticks. 

Epizootics is an outbreak of disease within animal populations at a high frequency. Moose, in particular, have been affected by winter ticks, leading to an increase in mortality rates among them. This epizootic — referred to as the winter-tick epizootic — is the subject of Beckford’s research. Beckford constructed a susceptible, infected model of tick epizootics in moose, representing moose afflicted by the ticks that cause them to experience anemia, weakness, starvation, and in extreme cases, death. This model is known as an SI Model, which examines the spread of infectious diseases and can predict their trajectories. 

Beckford explained that the model was created under the assumption that there are three seasons: fall, winter, and spring, given that the tick-moose relationship “had significant enough differences that warranted different systems of differential equations to model the susceptible and infected moose populations during these times.” In the autumn, the health of moose that harbor ticks are not significantly impacted. In the winter, the infestation of ticks ceases, “but moose which are infested with large tick loads have higher mortality than those which do not.” Beckford adds that “In the summer, there is a recovery of the infested moose individuals,” highlighting the notability of winter regarding the moose winter-tick epizootic. 

In this differential equation model, seasons are connected by pulses that condense the model by creating one pulse from the winter season to the next. Beckford explains, “the pulse from winter to summer describes birth since the relatively short birthing season of moose means that this can be considered as an instantaneous event rather than occurring throughout a season. The pulses from summer to autumn and autumn to winter are just linking these two systems of differential equations, making the ending point of the previous season the initial point of the next season.”

The model was initially designed as a three-season model with two pulses, which was later altered to have one season with one pulse. Given that this one season is winter, the pulse advances through the remainder of the year and to the beginning of the following winter. Beckford spoke on the importance of designing the model this way, stating, “It is important both analytically and for actually coding the model to run simulations.” Although it is possible to find points of equilibrium with a three-season model, the one-season winter model allows for a faster computation time and the ability to run a greater number of simulations. Over an extended period of time, simulations much like this one will make it easier to see how moose are affected by ticks and what measures can be taken to prevent harm to the moose population.

Beckford has recently completed the project’s analysis portion, which consisted of finding equilibrium points within the data and determining its stability. The next stage consists of parameter estimation, which is the application of data to the real world gathered from the model to figure out to what extent the natural environment can be simulated. 

This project’s results are especially significant in regard to the rapid rate at which climate change has made for shorter and warmer winters — which ticks favor — leading to increased complications in moose. Beckford states, “If our results are able to demonstrate how severely these tick epizootics affect moose, I think that it can be used as motivation to address climate change because [moose] are a token species, making way for national parks and different state initiatives to be enacted in order to protect moose.” 

Given that this endeavor required the application of mathematical concepts, both new and old, Beckford described online encyclopedias and various textbooks as “friends” during her research. When asked about difficulties that arose during research, Beckford states, “dropping a negative or a silly math mistake” has taught her the importance of checking her work often, as she exclaimed, “I learned that you should check your work, check it again, and when you think it is right, check it one more time.” 

Dr. Jasun Gong has been advising the project since its early stages and continues to be of assistance. Gong is currently an Associate Chair at Rose Hill, who has a Ph.D. in Mathematics from the University of Michigan. This past semester, Beckford also received help from David C. Elzinga, a graduate student from the University of Tennessee, studying mathematical psychology with the erudition of population dynamics and models much like that of the SI model used to simulate the moose winter-tick epizootic. Beckford noted that she is grateful for the help she has received thus far.

The application of mathematics to the environment does not stop here for Beckford. She has been accepted to numerous Ph.D. programs in applied mathematics, with one program offering a concentration in mathematical psychology. She hopes to pursue a career in which she can continue to relate math to the real world.

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